One way (of many) to solve this, is to calculate two values - $yesterday_start and $yesterday_end, and then check if the time stamp of the line in the log file lies between these two values. DateTime might prove useful. But beware of edge cases such as those days that switch from or to DST and other tricky business like that.

Another way could be to take $yesterday the way you do it now, and strip off the time information so that you only hold the date information. That's probably the easiest, because except for the way you obtaini $yesterday your code can run unaltered. But again, be careful that you don't run into problems on days that DST ends.

And finally, these days the three-arguments open (which you already use) with lexical filehandles is recommended. See perldoc open.